In our context, one generally use non-linear Young integration theory (see [1]} and [2]}) to link the space-time regularity of the averaged field (REF ) and the wellposedness of the integral equation (REF ). Nevertheless the example that our techniques will allow us to treat (Gaussian rough path, see Section ) will not allow us to go beyond \(b\in \mathcal {C}^\kappa \)  for \(\kappa >0\) . In that setting, Equation (REF ) always makes sense, always has a solution (thanks to Peano existence theorem) and one only to focus on uniqueness. The following Theorem is the main result in a general context in this setting :
